# What is the length of one side of the square

$a)$ In the figure, $ABCD$ is a square. $E$ and $F$ are the midmpoint of the sides $[AD]$ and $[CD]$ respectively. The are of the shaded region is $6cm^2$. What is the length of one side of the square.

My attemp is:$S_{\Delta ABG}=\frac{AB\cdot GG_1}{2},$ where $GG_1\perp AB$

$\Rightarrow$ $S_{\Delta ABG}=\frac{AB\cdot BC}{6},$ becouse i soppose that $GG_1=\frac{BC}{3},$ but I didint now it is correct. Now we have: $AB=6$

$$A = \dfrac{3x^2}{2} = 6$$
So $x = 2$
Therefore side length $S = 3x = 6$
From $G$ drop perpendicular on $AB$. Name it $D$.
As $\angle GDB=45^\circ$ , $GD=AD$. $$\therefore {GD\over AB-AD}={GD\over AB-GD}={EA\over AB}={1\over 2}\\\implies GD={AB\over3}$$ Now given $${1\over 2}AB\times GD=6\implies AB=6$$